Let 1 = a1 < a2 < a3 < … < ak = n be the positive divisors of n in increasing order.  For example, if n = 12, we have a1 = 1, a2 = 2, a3 = 3, a4 = 4, a5 = 6, a7 = 12. If n = (a3)3 – (a2)3, what is n?

Q: Prove that if n is a multiple of 4, then Fn is a multiple of 3. &#160;

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Q: Let n be an integer. Prove that if 2n3 +4 is divisible by 4, then n is even.

A: n is an integer. To show: 2n3 + 4 is divisible by 4 then n is even.

Q: Prove that vn if n is divisible by 3 then n2+2 is not divisible by 3

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Q: 5.2.8 suppose that a store offers gift certificates and denominations of $25 and $40. Determine the…

A: The given denominations are $25 and $40.Here, the given denominations are multiples of 5.The amounts…

Q: The generators of Z12 are: O {1,5,7,8,11} O {1,5,7,11} O {1,3,7,11} O {1,3,5,7,11}

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Q: Prove that for all integers n.4n² +6n + 1 is not divisible by 4. Be sure to explicitly show how each…

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Q: The generators of Z12 are: O {1,3,7,11} O {1,5,7,11} O {1,3,5,7,11} O {1,5,7,8,11} O O O O

A: Generators of Zn are 'a' such that: gcd (a ,n)= 1.

Q: Prove that (1 + x)n≥ (1 + nx), for all natural number n, where x &gt; – 1.

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Q: Prove that 4 < n! for all n 2 10 by induction

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Q: Prove that 3^1 + 3^2 + 3^3 + 3^4 + ... + 3^n = (3^(n+1) -3)/(2) for every n ∈ N

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Q: Prove by induction that (2n+1)3 is an odd number for all positiveintegers n

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Q: 4. Let, a, = 1, a2 = 2, b, = 0, b2 = 1, for n2 3, an = 2an-1+ bn-1 for n22 b, = bn-1+ an-1 express…

A: Given- Let a1 = 1, a2 = 2, b1 = 0, b2 = 1 for n ≥ 3,an = 2an-1 + bn-1for n ≥ 2,bn = bn-1 + an-1 To…

Q: -ん 2 kodd

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Q: 4. Let, a, = 1, az = 2, b, = 0, b2 = 1, for n 2 3, an = 2an-1+ bn-1 for n 22 bn = bn-1+ an-1 express…

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Q: Show that if a|m and a|(ms+nt) for some integers a≠0, m, s, n, and t, then a|nt.

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Q: 3. Prove that if a > 1, then a"/n → o.

A: We used binomial inequality

Q: If you have n random real number in a ordered list {a_1,a_2,…,a_n} , what is the probability of…

A: Given n random real number in a ordered list {a_1,a_2....a_n} {a_1,...a_i} is a monotonically…

Q: # 2. Use mathematical induction II to prove that for all integers n 2 5, 1 + 4n < 2".

A: Given: n≥5 To prove: 1+4n&lt;2n using mathematical induction.

Q: 4. Let, a1 = 3, a2 = 4 and for n ≥ 3, an = 2an-1 + an-2 + n2 , express an in terms of n.

A: As per guidelines we should able to do only one question at a time because both questions are…

Q: Let i be a positive integer and A, = [-i+1,1]. Then U2, A

A: We have to use properties of union in this question.

Q: Let fn be the n-th Fibonacci number of Example 3c in 8.1, k=n k=n Σ'=1+2+...+ B₁=k²= 1² +2²+. k=1…

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Q: Prove that there is some positive integer n such that n, n + 1,n + 2,... , n + 200 are all…

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Q: 2. Let f and g be any permutations. Show that the order of h = fgf- is the same as the order of g.

A: Here f and g are tow permutations  and we are given that h=fgf-1. We have to show that both f and g…

Q: 1. Show for all natural numbers n that (2k – 1) = 1+3+ ...+ (2n – 1) = n². k=1

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Q: Prove that there exists an n ∈ N such that 7 | (3^n − 1).

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Q: Let r(n) be the number of n-permutations whose square is the identity permutation. Prove that if n >…

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Q: Show that for any r, n > 1, п! Σ = r". nį!n2!...n,! nį+n2+…+n,=n, n¡ 20 11 (

A: Given information: The sample size is n. The number of categories is r.

Q: Show that there are infinitely many integers n such that 4n^2+1 is simultaneously divisible by 13…

A: We need to prove there are infinitely many integers n such that 4n^2+1 issimultaneously divisible by…

Q: 3. Prove that 6|(9" - 3") for all n E Z+.

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Q: Show that there are infinitely many integers n such that 4(n^2)+1 is simultaneously divisible by 13…

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Q: 3. Prove that when you select n+1 numbers from {1,2,...2n} two of them Must be consecutive (that is,…

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Q: . show that for every positive integer n, if n^2+2 is prime then n is a multuple of 3. Is the…

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Q: Prove that n3+(n+1)3+(n+2)3 is divisible by 9 for all non-negative integers  n.

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Q: Prove gcd(n, n+1) = 1.

A: To prove the below expression. gcdn, n+1=1

Q: Show that, for n > 1 2n 1.3.5... (2n – 1),2n 2.4.6...2n n

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Q: Let, a1 = 1, a2 = 2, b1 = 0, b2 = 1, for n ≥ 3, an = 2an-1+ bn-1 for n ≥ 2 bn = bn-1 + an-1…

A: The Fibonacci sequence can be defined as a series where a term of the sequence is the sum of the…

Q: Prove that a selection of 7 numbers from the set {1, 2, . . . , 11} contains numbers m and n such…

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Q: 8. (Chapter 10, #16) Prove that for every positive integer n, 2" +1< 3".

A: Using induction to prove any statement includes first proving the statement is true for n=1, then…

Q: Prove that Vn if n is divisible by 3 then n2+2 is not divisible by 3

A: If a and b are two integers. Suppose n is non zero integer such that |n|≠1. If a is divisible by n…

Q: Prove that if k <n and (a^k - b^k) | (a^n - b^n), then k |n

A: We will prove it using contradiction

Q: Let a1 = a2 = 1 and suppose an = an-1 + an-2 for n ≥ 3. Prove that a1a2 + a2a3 + · · · + a2n-1a2n =…

A: We have given that , a1 = a2 = 1 Also , an = an-1 + an-2 We need to prove that ,  a1a2 + a2a3 +…

Q: Prove that ℤ/n has exactly n elements.

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Q: If n is a positive integer, how many 5-tuples of integers from 1 through n can be formed in which…

A: The number of r combinations which can be selected from n numbers if repetitions allowed is…

Q: Consider all n! permutations of the set {1,2,3,..., n}. We say that a number i is a fixed point of a…

A: Given: pn(k) be number of permutations of 1, 2, ..., n that has exactly k fixed points.

Q: 5. Prove that for any integer n, at least one of the integers n, n+2, n +4 is divisible by 3.

A: According to guidelines we can solved only ist question thank you   we have to Prove that for any…

Q: How many permutations of{1,2, . . . ,8}are there in which no even integer is a fixed point

A: We will permutations to find our required result.

Q: 6. Prove that for n> 2, 2(;) + (*) =n². п

A: We have too prove the given expression.   According to our guidelines we can solve only one question…